대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제38권3호
  • /
  • Pages.733-740
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  • 2023
  • /
  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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THE DIMENSION GRAPH FOR MODULES OVER COMMUTATIVE RINGS

  • Shiroyeh Payrovi (Department of Mathematics Imam Khomeini International University)
  • 투고 : 2022.09.10
  • 심사 : 2023.02.07
  • 발행 : 2023.07.31
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초록

Let R be a commutative ring and M be an R-module. The dimension graph of M, denoted by DG(M), is a simple undirected graph whose vertex set is Z(M) ⧵ Ann(M) and two distinct vertices x and y are adjacent if and only if dim M/(x, y)M = min{dim M/xM, dim M/yM}. It is shown that DG(M) is a disconnected graph if and only if (i) Ass(M) = {𝖕, 𝖖}, Z(M) = 𝖕 ∪ 𝖖 and Ann(M) = 𝖕 ∩ 𝖖. (ii) dim M = dim R/𝖕 = dim R/𝖖. (iii) dim M/xM = dim M for all x ∈ Z(M) ⧵ Ann(M). Furthermore, it is shown that diam(DG(M)) ≤ 2 and gr(DG(M)) = 3, whenever M is Noetherian with |Z(M) ⧵ Ann(M)| ≥ 3 and DG(M) is a connected graph.

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참고문헌

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